vol III Development:
Chapter 1: Epistemology
page 2: Abstraction
Epistemology is quality control for knowledge, separating the true and trustworthy from the false and unreliable. Before we can talk about quality control, however, we need a theory of knowledge. This we base around the age old idea of abstraction. When I come to know something, like a bag of flour, I have something new inside me, not the bag of flour, but an abstract representation of the bag of flour.
On this site, we affirm the hypothesis proposed by Landauer, that all information is physically represented. Nevertheless, mathematics, by abstracting from physical embodiment, clarifies our view of the world and simplifies the verification of relationships between statements and reality. Landauer
It is this abstract representation of the bag of flour, encoded physically within me in some complex set of neural symbols, which enables me to imagine all sorts of things about that bag of flour: how to approach it, pick it up and load it into a truck; how to sample its quality; estimate how much to pay for it; foresee the pleasure of cooking with it and eating the result, and so on. Flour - Wikipedia
Mathematics
We have described scientific method as a cycle of imagination and testing. Modern science began its career when Galileo and others began to use mathematics to expand their imagined models of the world. Galileo: Il Saggiatore, Popper
Here we think of mathematics as the exploration of pure abstraction, that is forms not embodied in matter. Mathematicians can communicate anything that can be written down. Already we see that pure formalism is an unattainable ideal, because even writing is matter. Like the mental image of the bag of flour, however, mathematics cuts the matter content to a minimum by using meaning or correspondence. Formalism (mathematics) - Wikipedia
There is simply not enough matter in the Earth to construct a physical infinite set. Yet abstraction allows us to imagine, talk, and write about infinite sets. The practical criterion for mathematics is that its creations be consistent, useful and beautiful, capturing the essence of a structure as my imagination can capture the essence of a bag of flour. Imagination - Wikipedia
Mathematical language is an aid to consistency and communication. It is an extension of natural language. Mathematics uses numbers stretching to infinity to enable us to deal with huge sets of objects like all the points in a space, and complex relationships between them.
We see mathematics stretching back to the beginning of recorded history, so we can only speculate about its origins. Here we use naming as a starting point to explore the nature of mathematics. Naming establishes a correspondence between two things, the name and what is named. Kramer, Genesis 2:19-20
In natural languages, names are seen as very different from things. My name is a word; I am a massive and complex physical object. Mathematics deals only with names, so that its correspondences exist between names only. It ignores the physical embodiment of names that Landauer suggests is necessary for their realization.
Arithmetic and geometry
Traditionally, mathematics is divided into two areas, arithmetic and geometry. Arithmetic is concerned with numbers, geometry with shapes, forms, pictures and spaces. Applied arithmetic uses numbers to model the relationships of distinct, countable objects like sheep and monetary units. Applied geometry deals with the measurement and calculation of continuous objects like land and buildings. Arithmetic - Wikipedia, Geometry - Wikipedia
The interface between arithmetic and geometry is a fertile source of mathematics. It was here that people realized that integers were not enough to describe continuous geometric objects. So it become necessary to invent fractions (rational numbers). Then it was discovered that the diagonal of a unit square cannot be represented by a rational number. This pointed to the new world of real numbers which both includes the rational numbers, and fills the spaces between them. Heath
In addition to numbers, arithmetic is built on a set of operations on numbers, addition, subtraction, multiplication and division. These operations were put on a firm foundation with the invention of set theory by nineteenth century mathematician Georg Cantor. Using the imagery of sets, we can explain clearly to ourselves how things like addition and multiplication work. Dauben
Order and correspondence
Two important ideas in set theory are order and correspondence. The discovery of algebra led to the development of the complex numbers so that every algebraic equation would have a numerical solution. A complex number is an ordered pair of numbers with special rules for their arithmetic operations. Set theory does not restrict itself to ordered pairs, but may consider infinite ordered sets and operations that operate on such sets. This mathematical formalism has proved very useful in physics. Vector space - Wikipedia
Set theory put mathematics on such a firm foundation that Cantor discovered a new realm of numbers. Cantor's work was inspired by the need for an arithmetic treatment of the geometrical continuum.
He was able to reveal an enormously complex structure within the continuum which can be represented by transfinite numbers. The transfinite cardinal and ordinal numbers that Cantor invented form the mathematical backbone of this site. Cantor
Cantor imagined (and it seems true) that anything thinkable can be represented in the space of transfinite numbers. Following his lead, we hope to exploit transfinite numbers to model the observable elements of God.
Consistency, completeness and computability
As with most great inventions, set theory raised more questions than it answered. In the late nineteenth century, people were inclined to think that mathematics was a logical linguistic structure in which all questions could be answered. In other words, mathematics, although infinite, was in some way bounded.
Kurt Gödel and Alan Turing, building on Cantor's work, showed that this was not the case. Gödel found that if mathematics is consistent, it is not complete. Turing found that if mathematics is consistent, it is not computable. Kurt Gödel, Hodges
These two ideas are of great importance for theology, since they suggest that any God that appears consistent cannot be of a fixed infinity, but must be an ever growing, living and evolving entity. Such an abstract system could possibly describe our growing, living and evolving Universe, and our infinitely imaginative minds.
From the abstract mathematical point of view, then, there are no limits to a the meaning of a bag of flour, or anything else in the Universe. One can imagine that the creativity of our own imaginations is an image of the creativity of the universal imagination.
(revised 7 August 2014)